A Compressed Sampling Receiver Based on Modulated Wideband Converter and a Parameter Estimation Algorithm for Fractional Bandlimited LFM Signals


A modified modulated wideband converter (MWC)-based compressed sampling receiver and a parameter estimation algorithm in discrete time fractional Fourier domain (DTFrFD) to intercept and estimate the fractional bandlimited LFM signals are suggested in this work. Our proposed compressed sampling receiver has some advantages compared to the original MWC-based compressed sampling receiver. Firstly, the proposed receiver works in DTFrFD which is more effective to intercept the fractional bandlimited signals, while the original MWC-based digital receiver works in discrete time Fourier domain (DTFD). Secondly, the cross-channel signal can get better separation by the proposed digital receiver than by original MWC-based receiver because that the nonzero part of the signal’s available spectrum is narrower in DTFrFD than in DTFD. Then, with the data acquired from the digital receiver, we propose a novel parameter estimation algorithm for the intercepted fractional bandlimited LFM signal based on OMP and the differentiation spectrum in DTFrFD. The initial frequency and final frequency of the intercepted fractional bandlimited LFM signal can be extracted successfully without two-dimensional search and reconstruction, and only two iterations are needed according the algorithm. Both theoretical analysis and simulation results demonstrate that the proposed algorithm bears a relatively low complexity and its estimation precision is not only higher than both DFrFT-based search algorithm and DFrFT-based reconstruction algorithm but also close to Cramer–Rao lower bound.

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Modulated wideband converter


Digital fractional Fourier transform


Linear frequency modulation


Discrete time fractional Fourier domain


Orthogonal matching pursuit


Discrete time fractional Fourier transform


Discrete time Fourier transform


Discrete time Fourier domain


Mean squared error


Low probability of intercept


Signal-to-noise ratio


Fractional Fourier transform


Fractional Fourier domain


Simplified fractional Fourier transform


Fourier transform


Digital simplified fractional Fourier transform


Simplified fractional Fourier domain


Additive white Gaussian noise


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This research was funded by National Natural Science Foundation of China Grant No. 61271354 and Henan Province Science and Technology Key Project Grant No. 142102210431.

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Appendix A

Appendix A

Spectrum distribution characteristics of LFM signal in SFrFD A monocomponent LFM signal is defined by:


where \(f_0\) is the initial frequency. A is the amplitude of \(x\left( t\right) \) which could berandom or fixed. \(K_{\mathrm{lfm}}\) is the modulation rate. And the duration time of \(x\left( t\right) \) is \( \left[ -\frac{T_d}{2},\frac{T_d}{2}\right] \).

The SFrFT of \(x\left( t\right) \) can be calculated by


According to (S2), \({\overline{X}}_\alpha \left( u\right) \), the SFrFT of \( x\left( u\right) \), denotes the FT of another LFM signal whose modulation rate is \( K'_{\mathrm{lfm}} =K_{\mathrm{lfm}}+\cot \alpha /2 \pi \), initial frequency is \(f_0\), and the frequency interval is \(\left[ f_0-K'_{\mathrm{lfm}}\frac{T_d}{2}, f_0+K'_{\mathrm{lfm}}\frac{T_d}{2}\right] \). So the bandwidth is \(B'=K'_{\mathrm{lfm}}\cdot T_d \). It can be interpreted that the SFrFT rotates the time-frequency distribution curve of \( x\left( t\right) \) in the clockwise direction with angle \(\alpha \), so that the modulation rate and bandwidth changes.


$$\begin{aligned} {\overline{X}}_\alpha \left( u\right)= & {} \frac{A}{\sqrt{2K'_{\mathrm{lfm}}}}\exp \left( -\frac{j\left( u-2\pi u_0\csc \alpha \right) ^2}{4\pi K'_{\mathrm{lfm}}}\right) \cdot \left\{ \left[ c\left( u_2\right) \right. \right. \\&\left. \left. + c\left( u_1\right) \right] + j\left[ s\left( u_2\right) + s\left( u_1\right) \right] \right\} \end{aligned}$$

where \( c\left( u\right) = \int _{0}^{u} \cos \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x\) and \( s\left( u\right) = \int _{0}^{u} \sin \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x\) is the Fresnel integral. And


where \( u_0=f_0\sin \alpha \). So the amplitude spectrum of \({\overline{X}}_\alpha \left( u\right) \) is


and the phase spectrum is


Substituting \( K'_{\mathrm{lfm}}=\frac{B'}{T}\) into Eq. (S5):


According to the property associated with Fresnel integral, when \(B'T_d \gg 1\), i.e., \(K'_{\mathrm{lfm}}T_d^2 \gg 1\), \(c\left( u_2\right) \approx s\left( u_2\right) \approx \frac{1}{2}\), so

$$\begin{aligned} |{{\overline{X}}}_\alpha \left( u\right) |\approx & {} \frac{A}{\sqrt{K'_{\mathrm{lfm}}}}\cdot rect\left( \frac{u-2\pi u_0\csc \alpha }{B'}\right) \\= & {} \frac{A}{\sqrt{K_{\mathrm{lfm}}+\cot \alpha /2 \pi }}\cdot rect\left( \frac{u-2\pi u_0\csc \alpha }{B'}\right) ,\theta \left( u\right) \\\approx & {} - \frac{\pi \left( u-2\pi u_0\csc \alpha \right) ^2}{K'_{\mathrm{lfm}}}+\frac{\pi }{4}=- \frac{\pi \left( u-2\pi u_0\csc \alpha \right) ^2}{K_{\mathrm{lfm}}+\cot \alpha /2 \pi }+\frac{\pi }{4}. \end{aligned}$$

Therefore, when \(\left( K_{\mathrm{lfm}}+\cot \alpha /2 \pi \right) T_d^2\gg 1\), \(|{{\overline{X}}}_\alpha \left( u\right) |\) is approximately a rectangular spectrum.

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Li, X., Wang, H. & Luo, H. A Compressed Sampling Receiver Based on Modulated Wideband Converter and a Parameter Estimation Algorithm for Fractional Bandlimited LFM Signals. Circuits Syst Signal Process 40, 918–957 (2021). https://doi.org/10.1007/s00034-020-01507-6

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  • Modulated wideband converter
  • Fractional Fourier transform
  • Linear frequency modulation
  • Parameter estimation
  • Orthogonal matching pursuit
  • Cramer–Rao lower bound